Given an array of positive integers and a number S, find the longest contiguous subarray having the sum at most S.
Return the start and end indices denoting this subarray.
Example
Input: nums = [3, 2, 5, 2, 2, 1, 1, 3, 1 , 2]
, S = 11
Output: [3, 8]
Explanation:the subarray nums[3...8] of sum 10
Your algorithm should run in O(n^2) time and use O(1) extra space.
The problem requires finding the longest contiguous subarray within a given array of positive integers such that the sum of the subarray is at most a given number S. The output should be the start and end indices of this subarray.
nums
.S
.Input: nums = [3, 2, 5, 2, 2, 1, 1, 3, 1, 2], S = 11 Output: [3, 8] Explanation: The subarray nums[3...8] has a sum of 10, which is the longest subarray with sum at most 11.
The core challenge is to find the longest contiguous subarray whose sum does not exceed a given value S. This problem is significant in various applications such as resource allocation, budgeting, and data analysis where constraints must be respected.
Potential pitfalls include misunderstanding the requirement for the subarray to be contiguous and not considering all possible subarrays.
To solve this problem, we can use a sliding window approach to maintain a window of elements that have a sum less than or equal to S. This approach ensures that we efficiently find the longest subarray without having to check all possible subarrays explicitly.
A naive solution would involve checking all possible subarrays and calculating their sums. This would have a time complexity of O(n^3) and is not optimal for large arrays.
We can optimize the solution using a sliding window approach:
start
and end
, both set to the beginning of the array.currentSum
to store the sum of the current window.end
pointer and adding the element at end
to currentSum
.currentSum
exceeds S, move the start
pointer to the right until currentSum
is less than or equal to S.Here is a step-by-step breakdown of the sliding window algorithm:
start
and end
to 0, currentSum
to 0, and variables to store the maximum length and corresponding indices.end
pointer.end
to currentSum
.currentSum
exceeds S, increment the start
pointer and subtract the element at start
from currentSum
.public class LongestSubarrayWithSumAtMostS {
public static int[] findLongestSubarray(int[] nums, int S) {
int start = 0, end = 0, currentSum = 0;
int maxLength = 0;
int[] result = new int[2];
while (end < nums.length) {
// Add the current element to the currentSum
currentSum += nums[end];
// While currentSum exceeds S, move the start pointer to the right
while (currentSum > S && start <= end) {
currentSum -= nums[start];
start++;
}
// Update the maximum length and result indices if needed
if (end - start + 1 > maxLength) {
maxLength = end - start + 1;
result[0] = start;
result[1] = end;
}
// Move the end pointer to the right
end++;
}
return result;
}
public static void main(String[] args) {
int[] nums = {3, 2, 5, 2, 2, 1, 1, 3, 1, 2};
int S = 11;
int[] result = findLongestSubarray(nums, S);
System.out.println("Start index: " + result[0] + ", End index: " + result[1]);
}
}
The time complexity of the sliding window approach is O(n) because each element is processed at most twice (once by the end
pointer and once by the start
pointer). The space complexity is O(1) as we are using a constant amount of extra space.
Potential edge cases include:
To test the solution comprehensively, consider the following test cases:
When approaching such problems, consider the following tips:
In this blog post, we discussed how to find the longest contiguous subarray with a sum at most S using a sliding window approach. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills in programming.
For further reading and practice, consider the following resources: